You can certainly do this with some of the broadcast server technologies. EVS (used in sports for instant replays) is probably way overkill, but a Grass Valley Profile or something similar that will allow you to edit a file while it is still being recorded and then combine the clips into one continuous playlist.
They are available for rent – Bexel comes to mind, I’m sure there are other sources. They’re not cheap however, and unless you’ve got the time to learn the system, you’ll probably need an operator. It sure would be easier to just watch the thing live and switch to some kind of filler video during the breaks…